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Homework 6

For this homework, you can submit up to 3 times.
There is no numerical questions in this homework, but
we provide one simple python script for Q6.3 for you to play
with. Download it form
here (right click, "save as").

Q6.1 \(\quad\) Projection into slice

In the text book ( chpater 13, Equation (13.6) ),
you have seen that the dynamcis
in the slice is related to dynamics in the full state
space by
\[
\hat{v}(\hat{x}) = v(\hat{x}) -
\frac{v(\hat{x})^\top \cdot t'}{t(\hat{x})^\top \cdot t'}
t(\hat{x})
\]
We can treat the above formula as a projection process:
projection from full state space to slice by matrix
\[
h(\hat{x}) = I -
\frac{t(\hat{x}) t'^\top}{t(\hat{x})^\top \cdot t'}
\]
Then the relation is just
\(\hat{v}(\hat{x}) = h(\hat{x}) v(\hat{x})\). Here, all vectors
are column vectors, and \(\top\) denotes the transpose of a column
vector giving you a row vector. Putting two matrices/vectors
together means the usual matrix product; for instance,
\(t(\hat{x}) t'^\top\) is the matrix product of a n-dimensional
column vector and a n-dimensional row vector giving you a [n x n]
matrix. The dot product above is used just to emphasize that
the outcome is a number.
Actually,
Which of the following is not correct about
projection \(h(\hat{x})\) ? Please think about
the physical meaning.

Q6.2 \(\quad\) Projection of Jacobian

At week 2, we learn the relation between Jacobian in the
full state space and that on the Poincare section
( chapter 4, equation (4.25) ):
\[
\hat{J}_{ij} = \left(
\delta_{ik} - \frac{v'_i \partial_k U'}{v'\cdot \partial U'}
\right) J_{kj}
\]
This transformation looks similar to \(h(\hat{x})\) in Q6.1,
right? This motivates us to think about the relation between Jacobian
in the full state space and that in the slice. Suppose flow
\(f(x, t)\) has continous symmetry \(g(\phi)\). After choosing a proper
slice, we reduce the dynamics into the slice. Choose one point in the
full state space \(x_1 = x(0)\), and evolve it for a period \(t\), we get
\(x_2 = f(x(0), t)\), denote the Jacobian associated with this period
as \(J(x_2, x_1)\). At the same time,
the dynamics in the slice also traces out a
corresponding trajectory from \(\hat{x}_1\) to \(\hat{x}_2\), with
\(x_1 = g(\phi_1)\hat{x}_1\) and \(x_2 = g(\phi_2) \hat{x}_2 \). Denote
the Jacobian in the slice assoiated with the same period as
\(\hat{J}(\hat{x}_2, \hat{x}_1)\). What is the correct relation between
\(J(x_2, x_1)\) and \(\hat{J}(\hat{x}_2, \hat{x}_1)\) ?

Hint: try to think
about the physical meaning of the following expressions. Also you can
write a simple code using two modes system to verify your statement.

So along as you know the itinerary sequence (symbolic
dynamics), you can calculate the initial condition.
Now, itinerary \(\overline{000001}\) denotes the symbolic dynamics
of a periodic orbit with length 6. Here, overline means repeating.
What is the corresponding initial condition of this orbit ?
(choose the smallest one in the periodic orbit)

Q6.4 \(\quad\) "Golden mean" pruned map (Chapter 14 Exercise 14.6)

The first figure at the bottom of this page shows a symmetric tent map
on the unit interval such that its highest point belongs to a 3-cycle.
The slopes of the two branches have the same magnitude. Please try to
find the slope. What is the critical value of this map, namely, the
image of the
critical point of this map ?

Q6.5 \(\quad\) "Golden mean" map -- continued

If the symbolic dynamics is such that for \(x < 1/2\) we use
symbol 0 and for \( x > 1/2 \) we use symbol 1, which one of
the following is not an admissible itinerary of a periodic orbit
in this map ?

\( \overline{1} \)

\( \overline{01}\)

\( \overline{001}\)

\( \overline{01011}\)

\( \overline{101110} \)

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